Shan M. Removable singularities for solutions of anisotropic parabolic equations

The thesis is devoted to the study of asymptotic behavior of solutions of divergent nonlinear anisotropic parabolic equations near a singular point and obtaining conditions of removability of singularities for them. This problem is complicated by the fact that a general qualitative theory for anisotropic elliptic and parabolic equations is not constructed. In spite of the fact that the exact form of the fundamental solution for such equations is unknown, the conditions for removability of singularity for the anisotropic parabolic equations and for such equations with the absorption and gradient absorption terms were established. These conditions were obtained by the method of precise pointwise estimates of solutions of type "nonlinear potential" which was proposed by I. V. Skrypnyk for elliptic divergent quasilinear equations and adapted in this thesis for anisotropic parabolic equations. Particular attention is paid to the established estimates of the type of Keller-Osserman. They have many uses, in this paper used to obtain the conditions for removability of singularities for the equations with absorption and gradient absorption, as well as to obtain inequalities of the Harnack type. The model cases of the equations are the anisotropic porous medium equation and the same equation with absorption and gradient absorption. The main difficulty lies in the fact that we consider the case when some part of anisotropic exponents can be less than 1 (singular case) and the other can be greater than 1 (degenerate case). These two cases are typically considered separately in the literature, the definitions of the solution are formulated separately for each case, the qualitative properties of the solutions are also proved separately even in the isotropic case. In the thesis we have found a universal approach in the study of the properties of solutions of the anisotropic porous medium equation which does not depend on the values of anisotropic exponents.